Completing the Square First

Calculus 2 · Axiom Academy

LESSON Completing the Square First Reshape a messy quadratic into a clean u^2 + a^2 form — the setup that unlocks trigonometric substitution. 1. Why the Quadratic Resists Trig Sub Trig substitution keys off the standard patterns and u^2 - a^2 — each one centered on a single squared variable. Graph y = x^2 + 6x + 13 and the obstacle is visible: its turning point does not sit on the y -axis. It is shifted left, so the expression can't be read as "(something centered) ^2 plus a constant" the way trig sub needs. Watch the algebra as geometry: the x^2 square and the 6x rectangle are reshaped into one big square. Split the 6x into two 3x strips, lay one along each side of the x^2 square, and a corner is left empty. Filling it adds 9 — so we add and subtract 9 to keep the value unchanged. — the side length added to each edge of the square. 3^2 = 9 — the area of the missing corner we add and subtract. 13 - 9 = 4 , the constant outside the square. 3. The Shift That Cleans the Form Completing the square is really a change of variable, u = x + 3 . Geometrically it slides the whole parabola left by 3 until its vertex lands on the axis. Watch the curve y = x^2 + 6x + 13 glide over to become y = u^2 + 4 — a quadratic centered at the origin, exactly the form trig substitution is built for. 4. The Right Triangle Behind the Substitution

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